Heat 2 Tabs of the oil in a 12 inch skillet over medium heat. Add garlic, stir one minute. Add pasta, toss to combine. Add broth, cook uncovered over medium heat or until liquid is nearly absorbed, stirring occasionally. Remove from heat, let cool 10 minutes.
Transfer to a large bowl. Meanwhile, using a spiral vegetable slicer, cut zucchini and yellow squash into strands; snip strands into shorter lengths.
Add to spaghetti and cool completely. Pull cheese into thin strands; chill until serving.
Into a small bowl toss together tomatoes, onion, parsley,kosher salt, pepper, remaining oil and vinegar. Add cheese to spaghetti mixture. Top with tomato mixture and walnuts.